A317099 Number of series-reduced planted achiral trees whose leaves span an initial interval of positive integers appearing with multiplicities an integer partition of n.

1, 3, 4, 9, 8, 19, 16, 35, 35, 54, 57, 113, 102, 155, 189, 279, 298, 447, 491, 702, 813, 1063, 1256, 1759, 1967, 2542, 3050, 3902, 4566, 5882, 6843, 8676, 10205, 12612, 14908, 18608, 21638, 26510, 31292, 38150, 44584, 54185, 63262, 76308, 89371, 106818, 124755

Offset

1,2

Comments

In these trees, achiral means that all branches directly under any given node that is not a leaf or a cover of leaves are equal, and series-reduced means that every node that is not a leaf or a cover of leaves has at least two branches.

Links

Table of n, a(n) for n=1..47.

Example

The a(4) = 9 trees:
  (1111), ((11)(11)), (((1)(1))((1)(1))), ((1)(1)(1)(1)),
  (1112),
  (1122), ((12)(12)),
  (1123),
  (1234).
The a(6) = 19 trees:
  (111111), ((111)(111)), (((1)(1)(1))((1)(1)(1))), ((11)(11)(11)), (((1)(1))((1)(1))((1)(1))), ((1)(1)(1)(1)(1)(1)),
  (111112),
  (111122), ((112)(112)),
  (111123),
  (111222), ((12)(12)(12)),
  (111223),
  (111234),
  (112233), ((123)(123)),
  (112234),
  (112345),
  (123456).

Mathematica

b[n_]:=1+Sum[b[n/d], {d, Rest[Divisors[n]]}];
a[n_]:=Sum[b[GCD@@Length/@Split[ptn]], {ptn, IntegerPartitions[n]}];
Array[a, 30]

Crossrefs

Cf. A001678, A003238, A052409, A052410, A067824, A167865, A168532, A214577, A289078, A294336, A316782, A317100.

Sequence in context: A106833 A217578 A280616 * A317715 A372121 A305551

Adjacent sequences: A317096 A317097 A317098 * A317100 A317101 A317102

Keyword

nonn

Author

Gus Wiseman, Aug 01 2018

Status

approved